Instead of thinking of a linear transformation as changing each vector a different way, you can think of a linear transformation as a change of basis. Instead of shifting the point, you’re shifting the entire coordinate system. For example, the vector <1,2,3> moves 1 unit in the first basis vector, 2 units in the second basis vector, and 3 units in the third basis vector. However, instead of the basis vectors being e1, e2, and e3, they’re now T(e1), T(e2), and T(e3). The point is still <1,2,3> but with respect to new axes, that have rotated/stretched. So, Tx tells you what happens to the point x if the axes are changed. Additionally, multiplying by the inverse of T can be thought of as shifting the coordinates back.